Notation for Heisenberg Hamiltonian(adsbygoogle = window.adsbygoogle || []).push({});

## H_{exch}=-\sum_{i<j}2J_{i,j}\vec{S}_i \cdot \vec{S}_j ##

where the sumation extends over all spins pair of the crystal lattice. Suppose we have chain with four sites and periodic boundary condition

## H_{exch}=-J_{1,2}\vec{S}_1 \cdot \vec{S}_2-J_{2,3}\vec{S}_2 \cdot \vec{S}_3-J_{3,4}\vec{S}_3 \cdot \vec{S}_4-J_{4,1}\vec{S}_4 \cdot \vec{S}_1 ##

I can't see that this is the same.

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# Confused with notation

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